{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "乐学偶得版权所有 lexueoude.com 公众号：乐学Fintech"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Population Y: \n",
    "Mean(Y)= $\\mu= \\frac{1}{N} \\sum_{i=1}^{N}Y_{i}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Var(Y)=$\\sigma^{2} = \\frac{1}{N} \\sum_{i=1}^{N}(Y_{i}-\\mu)^{2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sample X: \n",
    "Mean(X)= $ \\overline X= \\frac{1}{n} \\sum_{j=1}^{n}X_{j}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Var(X)=$s^{2} = \\frac{1}{n-1} \\sum_{j=1}^{n}(X_{j}-\\overline X)^{2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0         7\n",
      "1         4\n",
      "2         8\n",
      "3         5\n",
      "4         7\n",
      "5        10\n",
      "6         3\n",
      "7         7\n",
      "8         8\n",
      "9         5\n",
      "10        4\n",
      "11        8\n",
      "12        8\n",
      "13        3\n",
      "14        6\n",
      "15        5\n",
      "16        2\n",
      "17        8\n",
      "18        6\n",
      "19        2\n",
      "20        5\n",
      "21        1\n",
      "22       10\n",
      "23        6\n",
      "24        9\n",
      "25        1\n",
      "26       10\n",
      "27        3\n",
      "28        7\n",
      "29        4\n",
      "         ..\n",
      "99970    10\n",
      "99971     8\n",
      "99972     1\n",
      "99973     8\n",
      "99974     2\n",
      "99975     3\n",
      "99976     1\n",
      "99977     8\n",
      "99978     5\n",
      "99979     9\n",
      "99980     8\n",
      "99981     2\n",
      "99982     6\n",
      "99983     4\n",
      "99984    10\n",
      "99985     4\n",
      "99986     4\n",
      "99987     8\n",
      "99988     8\n",
      "99989     3\n",
      "99990     8\n",
      "99991     8\n",
      "99992     7\n",
      "99993     2\n",
      "99994     5\n",
      "99995     5\n",
      "99996     4\n",
      "99997     1\n",
      "99998     4\n",
      "99999     1\n",
      "Length: 100000, dtype: int64\n"
     ]
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "from IPython.core.pylabtools import figsize\n",
    "figsize(15,5)\n",
    "import pandas as pd \n",
    "import numpy as np\n",
    "\n",
    "np.random.seed(42)\n",
    "# The population N's size is 100000\n",
    "N=100000\n",
    "population = pd.Series(np.random.randint(1,11,N))\n",
    "print(population)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
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       "      <td>6</td>\n",
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       "      <td>5</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>7</td>\n",
       "      <td>10</td>\n",
       "      <td>2</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>6</td>\n",
       "      <td>4</td>\n",
       "      <td>4</td>\n",
       "      <td>7</td>\n",
       "      <td>1</td>\n",
       "      <td>7</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>7</td>\n",
       "      <td>2</td>\n",
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       "      <td>1</td>\n",
       "      <td>7</td>\n",
       "      <td>1</td>\n",
       "      <td>8</td>\n",
       "      <td>9</td>\n",
       "      <td>5</td>\n",
       "      <td>6</td>\n",
       "      <td>6</td>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>5</td>\n",
       "      <td>8</td>\n",
       "      <td>7</td>\n",
       "      <td>10</td>\n",
       "      <td>7</td>\n",
       "      <td>10</td>\n",
       "      <td>1</td>\n",
       "      <td>4</td>\n",
       "      <td>9</td>\n",
       "      <td>5</td>\n",
       "      <td>...</td>\n",
       "      <td>2</td>\n",
       "      <td>3</td>\n",
       "      <td>10</td>\n",
       "      <td>2</td>\n",
       "      <td>8</td>\n",
       "      <td>10</td>\n",
       "      <td>4</td>\n",
       "      <td>7</td>\n",
       "      <td>4</td>\n",
       "      <td>6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>10</td>\n",
       "      <td>6</td>\n",
       "      <td>6</td>\n",
       "      <td>5</td>\n",
       "      <td>5</td>\n",
       "      <td>3</td>\n",
       "      <td>4</td>\n",
       "      <td>2</td>\n",
       "      <td>1</td>\n",
       "      <td>7</td>\n",
       "      <td>...</td>\n",
       "      <td>6</td>\n",
       "      <td>2</td>\n",
       "      <td>7</td>\n",
       "      <td>6</td>\n",
       "      <td>8</td>\n",
       "      <td>4</td>\n",
       "      <td>10</td>\n",
       "      <td>1</td>\n",
       "      <td>9</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>8</td>\n",
       "      <td>3</td>\n",
       "      <td>9</td>\n",
       "      <td>2</td>\n",
       "      <td>3</td>\n",
       "      <td>2</td>\n",
       "      <td>6</td>\n",
       "      <td>5</td>\n",
       "      <td>5</td>\n",
       "      <td>4</td>\n",
       "      <td>...</td>\n",
       "      <td>6</td>\n",
       "      <td>7</td>\n",
       "      <td>5</td>\n",
       "      <td>9</td>\n",
       "      <td>7</td>\n",
       "      <td>5</td>\n",
       "      <td>9</td>\n",
       "      <td>5</td>\n",
       "      <td>2</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>10</td>\n",
       "      <td>6</td>\n",
       "      <td>2</td>\n",
       "      <td>9</td>\n",
       "      <td>8</td>\n",
       "      <td>6</td>\n",
       "      <td>6</td>\n",
       "      <td>7</td>\n",
       "      <td>2</td>\n",
       "      <td>6</td>\n",
       "      <td>...</td>\n",
       "      <td>8</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>5</td>\n",
       "      <td>8</td>\n",
       "      <td>8</td>\n",
       "      <td>3</td>\n",
       "      <td>4</td>\n",
       "      <td>10</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>9</td>\n",
       "      <td>5</td>\n",
       "      <td>4</td>\n",
       "      <td>9</td>\n",
       "      <td>9</td>\n",
       "      <td>10</td>\n",
       "      <td>7</td>\n",
       "      <td>1</td>\n",
       "      <td>6</td>\n",
       "      <td>2</td>\n",
       "      <td>...</td>\n",
       "      <td>7</td>\n",
       "      <td>1</td>\n",
       "      <td>4</td>\n",
       "      <td>9</td>\n",
       "      <td>3</td>\n",
       "      <td>4</td>\n",
       "      <td>3</td>\n",
       "      <td>10</td>\n",
       "      <td>2</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>7</td>\n",
       "      <td>2</td>\n",
       "      <td>4</td>\n",
       "      <td>10</td>\n",
       "      <td>3</td>\n",
       "      <td>9</td>\n",
       "      <td>9</td>\n",
       "      <td>1</td>\n",
       "      <td>6</td>\n",
       "      <td>2</td>\n",
       "      <td>...</td>\n",
       "      <td>6</td>\n",
       "      <td>3</td>\n",
       "      <td>4</td>\n",
       "      <td>4</td>\n",
       "      <td>8</td>\n",
       "      <td>7</td>\n",
       "      <td>8</td>\n",
       "      <td>1</td>\n",
       "      <td>3</td>\n",
       "      <td>7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>5</td>\n",
       "      <td>7</td>\n",
       "      <td>2</td>\n",
       "      <td>7</td>\n",
       "      <td>5</td>\n",
       "      <td>2</td>\n",
       "      <td>1</td>\n",
       "      <td>10</td>\n",
       "      <td>1</td>\n",
       "      <td>6</td>\n",
       "      <td>...</td>\n",
       "      <td>9</td>\n",
       "      <td>2</td>\n",
       "      <td>3</td>\n",
       "      <td>9</td>\n",
       "      <td>2</td>\n",
       "      <td>6</td>\n",
       "      <td>10</td>\n",
       "      <td>2</td>\n",
       "      <td>6</td>\n",
       "      <td>9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>7</td>\n",
       "      <td>1</td>\n",
       "      <td>4</td>\n",
       "      <td>3</td>\n",
       "      <td>6</td>\n",
       "      <td>5</td>\n",
       "      <td>8</td>\n",
       "      <td>5</td>\n",
       "      <td>3</td>\n",
       "      <td>2</td>\n",
       "      <td>...</td>\n",
       "      <td>6</td>\n",
       "      <td>6</td>\n",
       "      <td>7</td>\n",
       "      <td>4</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>9</td>\n",
       "      <td>8</td>\n",
       "      <td>6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>8</td>\n",
       "      <td>2</td>\n",
       "      <td>5</td>\n",
       "      <td>10</td>\n",
       "      <td>3</td>\n",
       "      <td>3</td>\n",
       "      <td>4</td>\n",
       "      <td>6</td>\n",
       "      <td>2</td>\n",
       "      <td>5</td>\n",
       "      <td>...</td>\n",
       "      <td>8</td>\n",
       "      <td>5</td>\n",
       "      <td>2</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>8</td>\n",
       "      <td>6</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>9</td>\n",
       "      <td>1</td>\n",
       "      <td>2</td>\n",
       "      <td>6</td>\n",
       "      <td>2</td>\n",
       "      <td>4</td>\n",
       "      <td>7</td>\n",
       "      <td>3</td>\n",
       "      <td>9</td>\n",
       "      <td>10</td>\n",
       "      <td>...</td>\n",
       "      <td>2</td>\n",
       "      <td>5</td>\n",
       "      <td>10</td>\n",
       "      <td>10</td>\n",
       "      <td>10</td>\n",
       "      <td>7</td>\n",
       "      <td>8</td>\n",
       "      <td>2</td>\n",
       "      <td>10</td>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>8</td>\n",
       "      <td>7</td>\n",
       "      <td>3</td>\n",
       "      <td>2</td>\n",
       "      <td>3</td>\n",
       "      <td>10</td>\n",
       "      <td>10</td>\n",
       "      <td>3</td>\n",
       "      <td>7</td>\n",
       "      <td>7</td>\n",
       "      <td>...</td>\n",
       "      <td>7</td>\n",
       "      <td>9</td>\n",
       "      <td>4</td>\n",
       "      <td>5</td>\n",
       "      <td>6</td>\n",
       "      <td>3</td>\n",
       "      <td>10</td>\n",
       "      <td>7</td>\n",
       "      <td>2</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>30 rows × 500 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "    0    1    2    3    4    5    6    7    8    9   ...   490  491  492  493  \\\n",
       "0     3    4    1    3    3    7    6    8    6    9 ...     4    6    5    3   \n",
       "1     2    7    6    9    9    2   10    5   10   10 ...     5    7    9    6   \n",
       "2     2    1    5    3    7    3    4    4    7    7 ...     8    1    5    8   \n",
       "3     3   10    2    6    9    7   10    3    4    2 ...     4    3    2    4   \n",
       "4     1    3    5   10   10    5    9    2    3    3 ...     2    4   10    9   \n",
       "5     5    5    6    2    1    6    9    9    3    6 ...     9    1    9    9   \n",
       "6     6    2    1    3    1    9    4    2    1    4 ...    10    7    1   10   \n",
       "7     2   10    5    4    2    7    2    1    7    1 ...     4    2    4    3   \n",
       "8     1    3    5    9   10    1    1    3    6    6 ...     8    6    9   10   \n",
       "9    10    3    5    2    1    3    1    2    1    9 ...     5    1    1    7   \n",
       "10   10    1    8    8    1    3    5    2    7    5 ...     6    7    3    9   \n",
       "11    5    2    5    6    3    5    6    5    5    3 ...     9    1    7    9   \n",
       "12    5    9    7   10    7    9    1    6    8    8 ...     8    7    1    2   \n",
       "13    9    1    6    5    1    1    3    1    9    2 ...     9    1    9   10   \n",
       "14    4    3    2    2    6    5    9    7    1    8 ...     9    7    6    9   \n",
       "15    8    8    7    2    5    3    9    4    2    6 ...     5    8    3    5   \n",
       "16    4    7    2    6    3   10    3    6    6    3 ...     4    9    8    1   \n",
       "17    7    5    1    1   10    8    8    9    4    3 ...     9    6    4    5   \n",
       "18    6    4    4    7    1    7    1    1    7    2 ...     2    1    7    1   \n",
       "19    5    8    7   10    7   10    1    4    9    5 ...     2    3   10    2   \n",
       "20   10    6    6    5    5    3    4    2    1    7 ...     6    2    7    6   \n",
       "21    8    3    9    2    3    2    6    5    5    4 ...     6    7    5    9   \n",
       "22   10    6    2    9    8    6    6    7    2    6 ...     8    1    1    5   \n",
       "23    9    5    4    9    9   10    7    1    6    2 ...     7    1    4    9   \n",
       "24    7    2    4   10    3    9    9    1    6    2 ...     6    3    4    4   \n",
       "25    5    7    2    7    5    2    1   10    1    6 ...     9    2    3    9   \n",
       "26    7    1    4    3    6    5    8    5    3    2 ...     6    6    7    4   \n",
       "27    8    2    5   10    3    3    4    6    2    5 ...     8    5    2    6   \n",
       "28    9    1    2    6    2    4    7    3    9   10 ...     2    5   10   10   \n",
       "29    8    7    3    2    3   10   10    3    7    7 ...     7    9    4    5   \n",
       "\n",
       "    494  495  496  497  498  499  \n",
       "0     4    8    3    9   10    9  \n",
       "1     3    9    9    9    2    2  \n",
       "2     4   10    8    4    9   10  \n",
       "3     5    2    5    7    8    6  \n",
       "4     1    9    3    9    5    4  \n",
       "5     7   10    2    4    5    7  \n",
       "6     9    5    9   10    8    2  \n",
       "7     4    9    3    2    1    1  \n",
       "8     7    7    6   10    8    6  \n",
       "9     4    3    8    5    1    1  \n",
       "10    5    7   10    9    1    5  \n",
       "11    6    9    1    8    6    2  \n",
       "12    9    6    7   10    5    7  \n",
       "13    7   10    6    6    6    7  \n",
       "14    4    4    9    1    3    3  \n",
       "15    2    2    2    6    2    2  \n",
       "16    8   10    4    8    9    7  \n",
       "17    6    1    7   10    2   10  \n",
       "18    8    9    5    6    6    5  \n",
       "19    8   10    4    7    4    6  \n",
       "20    8    4   10    1    9    3  \n",
       "21    7    5    9    5    2   10  \n",
       "22    8    8    3    4   10    1  \n",
       "23    3    4    3   10    2    4  \n",
       "24    8    7    8    1    3    7  \n",
       "25    2    6   10    2    6    9  \n",
       "26    6    1    1    9    8    6  \n",
       "27    1    8    6    6    1    6  \n",
       "28   10    7    8    2   10    5  \n",
       "29    6    3   10    7    2    8  \n",
       "\n",
       "[30 rows x 500 columns]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "samples={}\n",
    "# The size of each sample\n",
    "n=30\n",
    "# We are going to draw 500 times of samples and each time ,we are going to take 30 of samples.\n",
    "num_of_samples= 500\n",
    "for i in range(num_of_samples):\n",
    "    samples[i]= population.sample(n).reset_index(drop=True)\n",
    "\n",
    "samples=pd.DataFrame(samples)\n",
    "samples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <th>2</th>\n",
       "      <td>4.632222</td>\n",
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       "      <th>3</th>\n",
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       "      <td>9.560000</td>\n",
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       "      <td>8.383333</td>\n",
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       "      <th>6</th>\n",
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       "      <th>9</th>\n",
       "      <td>6.823333</td>\n",
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       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>10.165556</td>\n",
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       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>9.440000</td>\n",
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       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>9.183333</td>\n",
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       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>7.266667</td>\n",
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       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>7.183333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>6.450000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>8.995556</td>\n",
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       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>8.898889</td>\n",
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       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>7.445556</td>\n",
       "    </tr>\n",
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       "      <th>20</th>\n",
       "      <td>7.405556</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>8.088889</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>8.088889</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>8.298889</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>9.023333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>9.050000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>7.623333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>5.248889</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>8.666667</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>6.422222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>470</th>\n",
       "      <td>6.195556</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>471</th>\n",
       "      <td>10.048889</td>\n",
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       "      <th>472</th>\n",
       "      <td>9.226667</td>\n",
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       "    <tr>\n",
       "      <th>473</th>\n",
       "      <td>6.160000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>474</th>\n",
       "      <td>7.912222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>475</th>\n",
       "      <td>8.693333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>476</th>\n",
       "      <td>6.315556</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>477</th>\n",
       "      <td>9.515556</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>478</th>\n",
       "      <td>6.890000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>479</th>\n",
       "      <td>9.645556</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>480</th>\n",
       "      <td>7.090000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>481</th>\n",
       "      <td>9.248889</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>482</th>\n",
       "      <td>9.128889</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>483</th>\n",
       "      <td>8.245556</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>484</th>\n",
       "      <td>6.672222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>485</th>\n",
       "      <td>4.848889</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>486</th>\n",
       "      <td>7.343333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>487</th>\n",
       "      <td>7.223333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>488</th>\n",
       "      <td>8.405556</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>489</th>\n",
       "      <td>8.626667</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>490</th>\n",
       "      <td>5.778889</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>491</th>\n",
       "      <td>7.410000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>492</th>\n",
       "      <td>8.688889</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>493</th>\n",
       "      <td>8.476667</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>494</th>\n",
       "      <td>6.022222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>495</th>\n",
       "      <td>8.312222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>496</th>\n",
       "      <td>8.498889</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>497</th>\n",
       "      <td>9.045556</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>498</th>\n",
       "      <td>9.648889</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>499</th>\n",
       "      <td>7.698889</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>500 rows × 1 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "             0\n",
       "0     7.965556\n",
       "1     7.782222\n",
       "2     4.632222\n",
       "3     9.410000\n",
       "4     9.560000\n",
       "5     8.383333\n",
       "6     9.848889\n",
       "7     6.778889\n",
       "8     7.595556\n",
       "9     6.823333\n",
       "10   10.165556\n",
       "11    9.440000\n",
       "12    9.183333\n",
       "13    8.395556\n",
       "14    7.266667\n",
       "15    7.183333\n",
       "16    6.450000\n",
       "17    8.995556\n",
       "18    8.898889\n",
       "19    7.445556\n",
       "20    7.405556\n",
       "21    8.088889\n",
       "22    8.088889\n",
       "23    8.298889\n",
       "24    9.023333\n",
       "25    9.050000\n",
       "26    7.623333\n",
       "27    5.248889\n",
       "28    8.666667\n",
       "29    6.422222\n",
       "..         ...\n",
       "470   6.195556\n",
       "471  10.048889\n",
       "472   9.226667\n",
       "473   6.160000\n",
       "474   7.912222\n",
       "475   8.693333\n",
       "476   6.315556\n",
       "477   9.515556\n",
       "478   6.890000\n",
       "479   9.645556\n",
       "480   7.090000\n",
       "481   9.248889\n",
       "482   9.128889\n",
       "483   8.245556\n",
       "484   6.672222\n",
       "485   4.848889\n",
       "486   7.343333\n",
       "487   7.223333\n",
       "488   8.405556\n",
       "489   8.626667\n",
       "490   5.778889\n",
       "491   7.410000\n",
       "492   8.688889\n",
       "493   8.476667\n",
       "494   6.022222\n",
       "495   8.312222\n",
       "496   8.498889\n",
       "497   9.045556\n",
       "498   9.648889\n",
       "499   7.698889\n",
       "\n",
       "[500 rows x 1 columns]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Delta degree of freedom ddof=0 diveded by n ddof=1 divided by n-1\n",
    "biased_samples=samples.var(ddof=0).to_frame()\n",
    "biased_samples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <td>7.965556</td>\n",
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       "      <th>1</th>\n",
       "      <td>7.873889</td>\n",
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       "      <th>2</th>\n",
       "      <td>6.793333</td>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>7.447500</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>7.870000</td>\n",
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       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>7.955556</td>\n",
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       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>8.226032</td>\n",
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       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>8.045139</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>7.995185</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>7.878000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>8.085960</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>8.198796</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>8.274530</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>8.283175</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>8.215407</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>8.150903</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>8.050850</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>8.103333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>8.145205</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>8.110222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>8.076667</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>8.077222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>8.077729</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>8.086944</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>8.124400</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>8.160000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>8.140123</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>8.036865</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>8.058582</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>8.004037</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>470</th>\n",
       "      <td>7.950528</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>471</th>\n",
       "      <td>7.954974</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>472</th>\n",
       "      <td>7.957663</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>473</th>\n",
       "      <td>7.953870</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>474</th>\n",
       "      <td>7.953782</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>475</th>\n",
       "      <td>7.955336</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>476</th>\n",
       "      <td>7.951898</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>477</th>\n",
       "      <td>7.955170</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>478</th>\n",
       "      <td>7.952946</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>479</th>\n",
       "      <td>7.956472</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>480</th>\n",
       "      <td>7.954671</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>481</th>\n",
       "      <td>7.957356</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>482</th>\n",
       "      <td>7.959781</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>483</th>\n",
       "      <td>7.960372</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>484</th>\n",
       "      <td>7.957716</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>485</th>\n",
       "      <td>7.951319</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>486</th>\n",
       "      <td>7.950071</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>487</th>\n",
       "      <td>7.948582</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>488</th>\n",
       "      <td>7.949516</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>489</th>\n",
       "      <td>7.950898</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>490</th>\n",
       "      <td>7.946474</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>491</th>\n",
       "      <td>7.945384</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>492</th>\n",
       "      <td>7.946892</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>493</th>\n",
       "      <td>7.947964</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>494</th>\n",
       "      <td>7.944074</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>495</th>\n",
       "      <td>7.944816</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>496</th>\n",
       "      <td>7.945931</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>497</th>\n",
       "      <td>7.948139</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>498</th>\n",
       "      <td>7.951548</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>499</th>\n",
       "      <td>7.951042</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>500 rows × 1 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "            0\n",
       "0    7.965556\n",
       "1    7.873889\n",
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       "3    7.447500\n",
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       "15   8.150903\n",
       "16   8.050850\n",
       "17   8.103333\n",
       "18   8.145205\n",
       "19   8.110222\n",
       "20   8.076667\n",
       "21   8.077222\n",
       "22   8.077729\n",
       "23   8.086944\n",
       "24   8.124400\n",
       "25   8.160000\n",
       "26   8.140123\n",
       "27   8.036865\n",
       "28   8.058582\n",
       "29   8.004037\n",
       "..        ...\n",
       "470  7.950528\n",
       "471  7.954974\n",
       "472  7.957663\n",
       "473  7.953870\n",
       "474  7.953782\n",
       "475  7.955336\n",
       "476  7.951898\n",
       "477  7.955170\n",
       "478  7.952946\n",
       "479  7.956472\n",
       "480  7.954671\n",
       "481  7.957356\n",
       "482  7.959781\n",
       "483  7.960372\n",
       "484  7.957716\n",
       "485  7.951319\n",
       "486  7.950071\n",
       "487  7.948582\n",
       "488  7.949516\n",
       "489  7.950898\n",
       "490  7.946474\n",
       "491  7.945384\n",
       "492  7.946892\n",
       "493  7.947964\n",
       "494  7.944074\n",
       "495  7.944816\n",
       "496  7.945931\n",
       "497  7.948139\n",
       "498  7.951548\n",
       "499  7.951042\n",
       "\n",
       "[500 rows x 1 columns]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "biased_samples=biased_samples.expanding().mean()\n",
    "biased_samples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>biased var estimate (divided by n)</th>\n",
       "    </tr>\n",
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       "  <tbody>\n",
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       "      <th>0</th>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>7.447500</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>7.870000</td>\n",
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       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>7.955556</td>\n",
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       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>8.226032</td>\n",
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       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>8.045139</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>7.995185</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>7.878000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>8.085960</td>\n",
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       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>8.198796</td>\n",
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       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>8.274530</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>8.283175</td>\n",
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       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>8.215407</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>8.150903</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>8.050850</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>8.103333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>8.145205</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>8.110222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>8.076667</td>\n",
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       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>8.077222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>8.077729</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>8.086944</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>8.124400</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>8.160000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>8.140123</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>8.036865</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>8.058582</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>8.004037</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>470</th>\n",
       "      <td>7.950528</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>471</th>\n",
       "      <td>7.954974</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>472</th>\n",
       "      <td>7.957663</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>473</th>\n",
       "      <td>7.953870</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>474</th>\n",
       "      <td>7.953782</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>475</th>\n",
       "      <td>7.955336</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>476</th>\n",
       "      <td>7.951898</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>477</th>\n",
       "      <td>7.955170</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>478</th>\n",
       "      <td>7.952946</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>479</th>\n",
       "      <td>7.956472</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>480</th>\n",
       "      <td>7.954671</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>481</th>\n",
       "      <td>7.957356</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>482</th>\n",
       "      <td>7.959781</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>483</th>\n",
       "      <td>7.960372</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>484</th>\n",
       "      <td>7.957716</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>485</th>\n",
       "      <td>7.951319</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>486</th>\n",
       "      <td>7.950071</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>487</th>\n",
       "      <td>7.948582</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>488</th>\n",
       "      <td>7.949516</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>489</th>\n",
       "      <td>7.950898</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>490</th>\n",
       "      <td>7.946474</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>491</th>\n",
       "      <td>7.945384</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>492</th>\n",
       "      <td>7.946892</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>493</th>\n",
       "      <td>7.947964</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>494</th>\n",
       "      <td>7.944074</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>495</th>\n",
       "      <td>7.944816</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>496</th>\n",
       "      <td>7.945931</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>497</th>\n",
       "      <td>7.948139</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>498</th>\n",
       "      <td>7.951548</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>499</th>\n",
       "      <td>7.951042</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>500 rows × 1 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "     biased var estimate (divided by n)\n",
       "0                              7.965556\n",
       "1                              7.873889\n",
       "2                              6.793333\n",
       "3                              7.447500\n",
       "4                              7.870000\n",
       "5                              7.955556\n",
       "6                              8.226032\n",
       "7                              8.045139\n",
       "8                              7.995185\n",
       "9                              7.878000\n",
       "10                             8.085960\n",
       "11                             8.198796\n",
       "12                             8.274530\n",
       "13                             8.283175\n",
       "14                             8.215407\n",
       "15                             8.150903\n",
       "16                             8.050850\n",
       "17                             8.103333\n",
       "18                             8.145205\n",
       "19                             8.110222\n",
       "20                             8.076667\n",
       "21                             8.077222\n",
       "22                             8.077729\n",
       "23                             8.086944\n",
       "24                             8.124400\n",
       "25                             8.160000\n",
       "26                             8.140123\n",
       "27                             8.036865\n",
       "28                             8.058582\n",
       "29                             8.004037\n",
       "..                                  ...\n",
       "470                            7.950528\n",
       "471                            7.954974\n",
       "472                            7.957663\n",
       "473                            7.953870\n",
       "474                            7.953782\n",
       "475                            7.955336\n",
       "476                            7.951898\n",
       "477                            7.955170\n",
       "478                            7.952946\n",
       "479                            7.956472\n",
       "480                            7.954671\n",
       "481                            7.957356\n",
       "482                            7.959781\n",
       "483                            7.960372\n",
       "484                            7.957716\n",
       "485                            7.951319\n",
       "486                            7.950071\n",
       "487                            7.948582\n",
       "488                            7.949516\n",
       "489                            7.950898\n",
       "490                            7.946474\n",
       "491                            7.945384\n",
       "492                            7.946892\n",
       "493                            7.947964\n",
       "494                            7.944074\n",
       "495                            7.944816\n",
       "496                            7.945931\n",
       "497                            7.948139\n",
       "498                            7.951548\n",
       "499                            7.951042\n",
       "\n",
       "[500 rows x 1 columns]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "biased_samples.columns=[\"biased var estimate (divided by n)\"]\n",
    "biased_samples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x112fca470>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "biased_samples.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>8.240230</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>8.050575</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>4.791954</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>9.734483</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>9.889655</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>8.672414</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>10.188506</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>7.012644</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>7.857471</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>7.058621</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>10.516092</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>9.765517</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>9.500000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>8.685057</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>7.517241</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>7.431034</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>6.672414</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>9.305747</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>9.205747</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>7.702299</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>7.660920</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>8.367816</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>8.367816</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>8.585057</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>9.334483</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>9.362069</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>7.886207</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>5.429885</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>8.965517</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>6.643678</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>470</th>\n",
       "      <td>6.409195</td>\n",
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       "    <tr>\n",
       "      <th>471</th>\n",
       "      <td>10.395402</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>472</th>\n",
       "      <td>9.544828</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>473</th>\n",
       "      <td>6.372414</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>474</th>\n",
       "      <td>8.185057</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>475</th>\n",
       "      <td>8.993103</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>476</th>\n",
       "      <td>6.533333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>477</th>\n",
       "      <td>9.843678</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>478</th>\n",
       "      <td>7.127586</td>\n",
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       "    <tr>\n",
       "      <th>479</th>\n",
       "      <td>9.978161</td>\n",
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       "    <tr>\n",
       "      <th>480</th>\n",
       "      <td>7.334483</td>\n",
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       "    <tr>\n",
       "      <th>481</th>\n",
       "      <td>9.567816</td>\n",
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       "    <tr>\n",
       "      <th>482</th>\n",
       "      <td>9.443678</td>\n",
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       "    <tr>\n",
       "      <th>483</th>\n",
       "      <td>8.529885</td>\n",
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       "    <tr>\n",
       "      <th>484</th>\n",
       "      <td>6.902299</td>\n",
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       "    <tr>\n",
       "      <th>485</th>\n",
       "      <td>5.016092</td>\n",
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       "    <tr>\n",
       "      <th>486</th>\n",
       "      <td>7.596552</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>487</th>\n",
       "      <td>7.472414</td>\n",
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       "    <tr>\n",
       "      <th>488</th>\n",
       "      <td>8.695402</td>\n",
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       "    <tr>\n",
       "      <th>489</th>\n",
       "      <td>8.924138</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>490</th>\n",
       "      <td>5.978161</td>\n",
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       "    <tr>\n",
       "      <th>491</th>\n",
       "      <td>7.665517</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>492</th>\n",
       "      <td>8.988506</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>493</th>\n",
       "      <td>8.768966</td>\n",
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       "    <tr>\n",
       "      <th>494</th>\n",
       "      <td>6.229885</td>\n",
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       "    <tr>\n",
       "      <th>495</th>\n",
       "      <td>8.598851</td>\n",
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       "    <tr>\n",
       "      <th>496</th>\n",
       "      <td>8.791954</td>\n",
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       "    <tr>\n",
       "      <th>497</th>\n",
       "      <td>9.357471</td>\n",
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       "    <tr>\n",
       "      <th>498</th>\n",
       "      <td>9.981609</td>\n",
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       "      <th>499</th>\n",
       "      <td>7.964368</td>\n",
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       "  </tbody>\n",
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       "<p>500 rows × 1 columns</p>\n",
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       "10   10.516092\n",
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       "12    9.500000\n",
       "13    8.685057\n",
       "14    7.517241\n",
       "15    7.431034\n",
       "16    6.672414\n",
       "17    9.305747\n",
       "18    9.205747\n",
       "19    7.702299\n",
       "20    7.660920\n",
       "21    8.367816\n",
       "22    8.367816\n",
       "23    8.585057\n",
       "24    9.334483\n",
       "25    9.362069\n",
       "26    7.886207\n",
       "27    5.429885\n",
       "28    8.965517\n",
       "29    6.643678\n",
       "..         ...\n",
       "470   6.409195\n",
       "471  10.395402\n",
       "472   9.544828\n",
       "473   6.372414\n",
       "474   8.185057\n",
       "475   8.993103\n",
       "476   6.533333\n",
       "477   9.843678\n",
       "478   7.127586\n",
       "479   9.978161\n",
       "480   7.334483\n",
       "481   9.567816\n",
       "482   9.443678\n",
       "483   8.529885\n",
       "484   6.902299\n",
       "485   5.016092\n",
       "486   7.596552\n",
       "487   7.472414\n",
       "488   8.695402\n",
       "489   8.924138\n",
       "490   5.978161\n",
       "491   7.665517\n",
       "492   8.988506\n",
       "493   8.768966\n",
       "494   6.229885\n",
       "495   8.598851\n",
       "496   8.791954\n",
       "497   9.357471\n",
       "498   9.981609\n",
       "499   7.964368\n",
       "\n",
       "[500 rows x 1 columns]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "unbiased_sample=samples.var(ddof=1).to_frame()\n",
    "unbiased_sample"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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       "      <th>10</th>\n",
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       "      <th>11</th>\n",
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       "      <th>14</th>\n",
       "      <td>8.498697</td>\n",
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       "      <th>15</th>\n",
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       "      <th>16</th>\n",
       "      <td>8.328465</td>\n",
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       "      <th>17</th>\n",
       "      <td>8.382759</td>\n",
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       "      <th>20</th>\n",
       "      <td>8.355172</td>\n",
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       "      <th>21</th>\n",
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       "      <th>22</th>\n",
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       "      <th>23</th>\n",
       "      <td>8.365805</td>\n",
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       "      <th>24</th>\n",
       "      <td>8.404552</td>\n",
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       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>8.441379</td>\n",
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       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>8.420817</td>\n",
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       "      <th>27</th>\n",
       "      <td>8.313998</td>\n",
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       "      <th>29</th>\n",
       "      <td>8.280038</td>\n",
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       "    <tr>\n",
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       "      <td>8.228142</td>\n",
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       "      <td>8.228051</td>\n",
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       "      <td>8.229658</td>\n",
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       "      <th>476</th>\n",
       "      <td>8.226102</td>\n",
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       "      <th>477</th>\n",
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       "      <td>8.232120</td>\n",
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       "      <td>8.224211</td>\n",
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       "      <td>8.222032</td>\n",
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       "      <th>494</th>\n",
       "      <td>8.218008</td>\n",
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       "      <td>8.218775</td>\n",
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       "      <td>8.219929</td>\n",
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       "      <td>8.222213</td>\n",
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       "      <td>8.225739</td>\n",
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       "      <td>8.225216</td>\n",
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       "26   8.420817\n",
       "27   8.313998\n",
       "28   8.336465\n",
       "29   8.280038\n",
       "..        ...\n",
       "470  8.224685\n",
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       "472  8.232065\n",
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       "493  8.222032\n",
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       "495  8.218775\n",
       "496  8.219929\n",
       "497  8.222213\n",
       "498  8.225739\n",
       "499  8.225216\n",
       "\n",
       "[500 rows x 1 columns]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "unbiased_sample=unbiased_sample.expanding().mean()\n",
    "unbiased_sample"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
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       "      <td>8.149655</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>8.364786</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>8.481513</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>8.559859</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>8.568801</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>8.498697</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>8.431968</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>8.328465</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>8.382759</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>8.426074</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>8.389885</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>8.355172</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>8.355747</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>8.356272</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>8.365805</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>8.404552</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>8.441379</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>8.420817</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>8.313998</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>8.336465</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>8.280038</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>470</th>\n",
       "      <td>8.224685</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>471</th>\n",
       "      <td>8.229284</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>472</th>\n",
       "      <td>8.232065</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>473</th>\n",
       "      <td>8.228142</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>474</th>\n",
       "      <td>8.228051</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>475</th>\n",
       "      <td>8.229658</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>476</th>\n",
       "      <td>8.226102</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>477</th>\n",
       "      <td>8.229486</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>478</th>\n",
       "      <td>8.227185</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>479</th>\n",
       "      <td>8.230833</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>480</th>\n",
       "      <td>8.228970</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>481</th>\n",
       "      <td>8.231748</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>482</th>\n",
       "      <td>8.234257</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>483</th>\n",
       "      <td>8.234867</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>484</th>\n",
       "      <td>8.232120</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>485</th>\n",
       "      <td>8.225503</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>486</th>\n",
       "      <td>8.224211</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>487</th>\n",
       "      <td>8.222671</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>488</th>\n",
       "      <td>8.223637</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>489</th>\n",
       "      <td>8.225067</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>490</th>\n",
       "      <td>8.220491</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>491</th>\n",
       "      <td>8.219363</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>492</th>\n",
       "      <td>8.220923</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>493</th>\n",
       "      <td>8.222032</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>494</th>\n",
       "      <td>8.218008</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>495</th>\n",
       "      <td>8.218775</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>496</th>\n",
       "      <td>8.219929</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>497</th>\n",
       "      <td>8.222213</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>498</th>\n",
       "      <td>8.225739</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>499</th>\n",
       "      <td>8.225216</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>500 rows × 1 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "     unbiased var estimate(divided by n-1)\n",
       "0                                 8.240230\n",
       "1                                 8.145402\n",
       "2                                 7.027586\n",
       "3                                 7.704310\n",
       "4                                 8.141379\n",
       "5                                 8.229885\n",
       "6                                 8.509688\n",
       "7                                 8.322557\n",
       "8                                 8.270881\n",
       "9                                 8.149655\n",
       "10                                8.364786\n",
       "11                                8.481513\n",
       "12                                8.559859\n",
       "13                                8.568801\n",
       "14                                8.498697\n",
       "15                                8.431968\n",
       "16                                8.328465\n",
       "17                                8.382759\n",
       "18                                8.426074\n",
       "19                                8.389885\n",
       "20                                8.355172\n",
       "21                                8.355747\n",
       "22                                8.356272\n",
       "23                                8.365805\n",
       "24                                8.404552\n",
       "25                                8.441379\n",
       "26                                8.420817\n",
       "27                                8.313998\n",
       "28                                8.336465\n",
       "29                                8.280038\n",
       "..                                     ...\n",
       "470                               8.224685\n",
       "471                               8.229284\n",
       "472                               8.232065\n",
       "473                               8.228142\n",
       "474                               8.228051\n",
       "475                               8.229658\n",
       "476                               8.226102\n",
       "477                               8.229486\n",
       "478                               8.227185\n",
       "479                               8.230833\n",
       "480                               8.228970\n",
       "481                               8.231748\n",
       "482                               8.234257\n",
       "483                               8.234867\n",
       "484                               8.232120\n",
       "485                               8.225503\n",
       "486                               8.224211\n",
       "487                               8.222671\n",
       "488                               8.223637\n",
       "489                               8.225067\n",
       "490                               8.220491\n",
       "491                               8.219363\n",
       "492                               8.220923\n",
       "493                               8.222032\n",
       "494                               8.218008\n",
       "495                               8.218775\n",
       "496                               8.219929\n",
       "497                               8.222213\n",
       "498                               8.225739\n",
       "499                               8.225216\n",
       "\n",
       "[500 rows x 1 columns]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "unbiased_sample.columns=[\"unbiased var estimate(divided by n-1)\"]\n",
    "unbiased_sample"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x11d1dada0>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax=unbiased_sample.plot()\n",
    "biased_samples.plot(ax=ax)\n",
    "real_population_variance=pd.Series(population.var(ddof=0),index=samples.columns)\n",
    "real_population_variance.plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "unbiased estimator:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{equation}\n",
    "    E(s^2) = E\\left(\\frac{1}{n - 1} \\sum_{j=1}^{n} (X_j - \\bar{X})^2\\right) = \\sigma^2\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "properties:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{equation}\n",
    "\\left\\{\n",
    "\\begin{aligned}\n",
    "    & E(Z_1 + Z_2) = E(Z_1) + E(Z_2), \\text{ for any } Z_1, Z_2 \\\\\n",
    "    & \\text{Var}(a Z) = a^2 \\text{Var}(Z), \\text{ for any } Z \\\\\n",
    "    & \\text{Var}(Z_1 + Z_2) = \\text{Var}(Z_1) + \\text{Var}(Z_2), \\text{ if } Z_1 \\text{ and } Z_2 \\text{ are independent} \\\\\n",
    "\\end{aligned}\n",
    "\\right.\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{equation}\n",
    "    \\text{Var}(Z) = E((Z - E(Z))^2) = E(Z^2 - 2ZE(Z) + E(Z)^2) = E(Z^2) - E(Z)^2\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{equation}\n",
    "    E(Z^2) = \\text{Var}(Z) + E(Z)^2\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{aligned}\n",
    "    E(s^2) = E\\left(\\frac{1}{n - 1} \\sum_{j=1}^{n} (X_j - \\bar{X})^2\\right) = & \\frac{1}{n - 1} E \\left( \\sum_{j=1}^{n} (X_j^2 - 2X_j\\bar{X} + \\bar{X}^2) \\right) \\\\\n",
    "    = & \\ \\frac{1}{n - 1} E \\left( \\sum_{j=1}^{n} X_j^2 - 2n\\bar{X}^2 + n\\bar{X}^2 \\right) \\\\\n",
    "    = & \\ \\frac{1}{n - 1} E \\left( \\sum_{j=1}^{n} X_j^2 - n\\bar{X}^2 \\right) \\\\\n",
    "    = & \\ \\frac{1}{n - 1} \\left[ E \\left( \\sum_{j=1}^{n} X_j^2 \\right) - E \\left( n\\bar{X}^2 \\right) \\right] \\\\\n",
    "    = & \\ \\frac{1}{n - 1} \\left[ \\sum_{j=1}^{n} E \\left( X_j^2 \\right) - n E \\left( \\bar{X}^2 \\right) \\right] \\\\\n",
    "\\end{aligned}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. First term:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{aligned}\n",
    "    \\sum_{j=1}^{n} E \\left( X_j^2 \\right) = & \\sum_{j=1}^{n} \\left( Var(X_j) + E(X_j)^2 \\right) \\\\\n",
    "    = & \\sum_{j=1}^{n} \\left( \\sigma^2 + \\mu ^2 \\right) \\\\\n",
    "    = & \\ n \\sigma^2 + n \\mu ^2 \\\\\n",
    "\\end{aligned}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2.Second term:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{aligned}\n",
    "    E(\\bar{X}^2) = & \\ Var(\\bar{X}) + E(\\bar{X})^2 \\\\\n",
    "    = & Var(\\frac{1}{n} \\sum_{j=1}^{n} X_j) + \\mu ^2 \\\\\n",
    "    = & \\frac{1}{n^2} Var(\\sum_{j=1}^{n} X_j) + \\mu ^2 \\\\\n",
    "    = & \\frac{1}{n^2} \\sum_{j=1}^{n} Var(X_j) + \\mu ^2, \\text{ because all } X_j\\text{'s are independent} \\\\\n",
    "    = & \\frac{1}{n^2} n\\sigma^2 + \\mu ^2 \\\\\n",
    "    = & \\frac{1}{n} \\sigma^2 + \\mu ^2 \\\\\n",
    "\\end{aligned}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{aligned}\n",
    "    E(s^2) = & \\ \\frac{1}{n-1} \\left[ \\sum_{j=1}^{n} E \\left( X_j^2 \\right) - n E \\left(\\bar{X}^2 \\right) \\right] \\\\\n",
    "    = & \\ \\frac{1}{n-1} \\left[n \\sigma^2 + n \\mu ^2 - n \\left( \\frac{1}{n} \\sigma^2 + \\mu ^2 \\right) \\right] \\\\\n",
    "    = & \\ \\frac{1}{n-1} \\left[n \\sigma^2 + n \\mu ^2 - \\sigma^2 - n \\mu ^2 \\right] \\\\\n",
    "    = & \\ \\sigma^2 \\\\\n",
    "\\end{aligned}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{equation}\n",
    "    \\sum_{j=1}^{n} (X_j - \\mu)^2 \\geq \\sum_{j=1}^{n} (X_j - \\bar{X})^2\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{aligned}\n",
    "    E\\left(\\sum_{j=1}^{n} (X_j - \\mu)^2\\right) = & \\ E \\left(\\sum_{j=1}^{n} (X_j - \\bar{X} + \\bar{X} - \\mu)^2\\right) \\\\\n",
    "    = & \\ E \\left(\\sum_{j=1}^{n} (X_j - \\bar{X})^2 + \\sum_{j=1}^{n} 2(X_j - \\bar{X})(\\bar{X} - \\mu) + \\sum_{j=1}^{n} (\\bar{X} - \\mu)^2 \\right) \\\\\n",
    "    = & \\ E \\left(\\sum_{j=1}^{n} (X_j - \\bar{X})^2 + \\sum_{j=1}^{n} (\\bar{X} - \\mu)^2 \\right) \\\\\n",
    "    = & \\ E \\left(\\sum_{j=1}^{n} (X_j - \\bar{X})^2 \\right) + E \\left(\\sum_{j=1}^{n} (\\bar{X} - \\mu)^2 \\right) \\\\\n",
    "    = & \\ E \\left(\\sum_{j=1}^{n} (X_j - \\bar{X})^2 \\right) + \\sum_{j=1}^{n} E \\left((\\bar{X} - \\mu)^2 \\right) \\\\\n",
    "    = & \\ E \\left(\\sum_{j=1}^{n} (X_j - \\bar{X})^2 \\right) + \\sum_{j=1}^{n} \\left( \\text{Var} (\\bar{X} - \\mu) + E (\\bar{X} - \\mu)^2 \\right) \\\\\n",
    "    = & \\ E \\left(\\sum_{j=1}^{n} (X_j - \\bar{X})^2 \\right) + \\sum_{j=1}^{n} \\left( \\text{Var} (\\bar{X}) + E (\\bar{X} - \\mu)^2 \\right) \\\\\n",
    "    = & \\ E \\left(\\sum_{j=1}^{n} (X_j - \\bar{X})^2 \\right) + \\sum_{j=1}^{n} \\text{Var} (\\bar{X}) \\\\\n",
    "    = & \\ E \\left(\\sum_{j=1}^{n} (X_j - \\bar{X})^2 \\right) + n \\text{Var} (\\bar{X}) \\\\\n",
    "    = & \\ E \\left(\\sum_{j=1}^{n} (X_j - \\bar{X})^2 \\right) + n \\text{Var} (\\frac{1}{n} \\sum_{j=1}^{n} X_j) \\\\\n",
    "    = & \\ E \\left(\\sum_{j=1}^{n} (X_j - \\bar{X})^2 \\right) + n \\frac{1}{n^2} \\sum_{j=1}^{n} \\text{Var} (X_j) \\\\\n",
    "    = & \\ E \\left(\\sum_{j=1}^{n} (X_j - \\bar{X})^2 \\right) + \\sigma^2 \\\\\n",
    "\\end{aligned}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{aligned}\n",
    "    E\\left(\\frac{1}{n} \\sum_{j=1}^{n} (X_j - \\mu)^2\\right) = & \\ \\frac{1}{n} E \\left(\\sum_{j=1}^{n} (X_j - \\mu)^2\\right) \\\\\n",
    "    = & \\ \\frac{1}{n} \\left[ E \\left(\\sum_{j=1}^{n} (X_j - \\bar{X})^2\\right) + \\sigma^2 \\right] \\\\\n",
    "        = & \\ \\frac{1}{n} \\left[ (n - 1) \\sigma^2 + \\sigma^2 \\right] \\\\\n",
    "    = & \\ \\sigma^2 \\\\\n",
    "\\end{aligned}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
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